Seismic wave propagation in nonlinear viscoelastic media using the auxiliary differential equation method

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Seismic wave propagation in nonlinear viscoelastic media using the auxiliary differential equation method

Roland Martin, Ludovic Bodet, Vincent Tournat, Fayçal Rejiba

To cite this version:

Roland Martin, Ludovic Bodet, Vincent Tournat, Fayçal Rejiba. Seismic wave propagation in non-

linear viscoelastic media using the auxiliary differential equation method. Geophysical Journal In-

ternational, Oxford University Press (OUP), 2019, 216 (1), pp.453-469. �10.1093/gji/ggy441�. �hal-

01980413�

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GJI Seismology

Seismic wave propagation in nonlinear viscoelastic media using the auxiliary differential equation method

Roland Martin,

¹

Ludovic Bodet,

²

Vincent Tournat

³

and Fayc¸al Rejiba

^2,*

1Laboratoire GET, Université Toulouse3Paul Sabatier, IRD, CNRS UMR5563, Observatoire Midi-Pyrénées,31400Toulouse, France. E-mail:

roland.martin@get.omp.eu

2Sorbonne Universit´e, CNRS, EPHE, METIS, F-75005Paris, France

3LAUM, UMR CNRS6613, Le Mans Universit´e, Avenue O. Messiaen, Le Mans, France

Accepted 2018 October 24. Received 2018 October 16; in original form 2018 March 19

S U M M A R Y

In previous studies, the auxiliary differential equation (ADE) method has been applied to 2-D seismic-wave propagation modelling in viscoelastic media. This method is based on the separation of the wave propagation equations derived from the constitutive law defining the stress–strain relation. We make here a 3-D extension of a finite-difference (FD) scheme to solve a system of separated equations consisting in the stress–strain rheological relation, the strain–velocity and the velocity–stress equations. The current 3-D FD scheme consists in the discretization of the second order formulation of a non-linear viscoelastic wave equation with a time actualization of the velocity and displacement fields. Compared to the usual memory variable formalism, the ADE method allows flexible implementation of complex expressions of the desired rheological model such as attenuation/viscoelastic models or even non-linear behaviours, with physical parameters that can be provided from dispersion analysis.

The method can also be associated with optimized perfectly matched layers-based boundary conditions that can be seen as additional attenuation (viscoelastic) terms. We present the results obtained for a non-linear viscoelastic model made of a Zener viscoelastic body associated with a non-linear quadratic strain term. Such non-linearity is relevant to define unconsolidated granular model behaviour. Thanks to a simple model, but without loss of generality, we demonstrate the accuracy of the proposed numerical approach.

Key words: Elasticity and anelasticity; Nonlinear differential equations; Numerical modelling; Computational seismology; Seismic attenuation; Wave propagation.

1 I N T R O D U C T I O N

In seismology or near surface geophysics applications, imaging and characterizing the Earth materials at different scales using seismic wave propagation can face attenuation and non-linear behaviour because of the physical nature of the media under study and the frequency content of the seismic sources. It can involve a wide variety of physical, geometrical, kinematical and structural types of non-linearities as well as combination of these non-linearities types (Ostrovsky & Johnson2001; Delsanto2006). Attenuation and non- linearity parametrizations in seismic modelling are not an easy task and are subject to many studies in the last two decades. Non-linear dynamics applied to geomaterials can be essentially due to the presence of soft features where damage is primarily observed as in granular media where the physical source of the non-linearities

∗Now at Normandie University, UNIROUEN, UNICAEN, CNRS, M2C, 76000 Rouen, France.

associated to grain-to-grain interactions has been well identified (Tournat & Gusev2010; Leglandet al.2012). At a larger scale, those non-linearities are arising during the wave propagation and are commonly studied in the context of site effect assessment, and related resonance phenomena or strong ground motion (de)amplification responses. In presence of non-linear site effects, viscous damping or non-linear ground behaviours have also been studied (Bonilla et al.2005,2011; R´egnieret al.2013, 2014, 2016b; Sandikkaya et al.2013; Akkaret al.2014). Actually, attenuation and non-linear characteristics of a medium are frequency dependent and are thus not easy to model. They are a consequence of complex mixture of superimposed distinct linear behaviour of each phase compos- ing the media. In practice, viscoelastic parameters are commonly estimated from seismic data and dispersion analysis, using empir- ical measurement of the quality factorQ, whether considered as quasi constant in the simpliest and common case over a specific frequency range or as frequency-dependent in case of dispersive and

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non-linear behaviour. But in many geophysical applications, this ap- proximation is too restrictive because the medium under study can have strong non-linear and attenuation behaviour depending on the seismic/acoustic signal amplitude and the frequency range. In the case of unconsolidated media (granular media, sedimentary basins, soften/weak fault systems), the parametrization of the attenuation and non-linearities must be done for different frequency ranges.

Several experiments done in this context have shown resonance effects and slow dynamics phenomena, with the appearance of harmonics in the frequency domain, that are evidences of attenuation and non-linearities dependent on frequency and strain amplitude.

As an example of application to earthquake dynamics, soften and weakened materials located at the seismic source, more particularly in the fault gouge area, can be at the origin of strong non-linearities and exhibit non-equilibrium and non-linear dynamics. Earthquake triggering can then be explained by fault failure caused by already weakened/jammed fault systems that are at critical and weak state and excited by seismic-waves impinging the fault system with a sufficiently high strain amplitude (Johnson & Jia2005). The seismic waves produced by a remote earthquake are exciting the fault and make the fault core modulus to decrease strongly, and the fault core weakens further. However, the strains should reach a sufficiently high threshold value to cause triggering phenomena. In such sit- uation, the stress–strain rheological relations characterizing these phenomena are generally non-linear and can be expressed as Taylor expansions of complex rheology expressions or high polynomial (at least cubic) functions of the strain close to the critical state.

Such non-linear rheologies can explain how some earthquakes can be triggered remotely by other huge earthquakes (Gomberget al.

2003; Gomberg & Johnson2005) like the 7.3Mw 1992 Landers earthquake (Hillet al.1993; Gomberget al.2001) or 7.1MwHec- tor Mine (Gomberg et al.2001) and 7.9Mw Denali earthquakes (Gomberget al.2004). In the context of the 2011 Tohoku earthquake, non-linear viscoelastic wave propagation modelling has been able to reproduce displacement amplifications in the near surface.

This has been enabled by adding non-linearities in both the elastic and attenuation parts of the stress tensor attenuation as well as hysteretic effects (d’Avilaet al. 2013) in the ground response due to strong seismic motion. As another example, several harmonics appearing in the observed accelerations spectra related to the Northridge earthquake (Del´epineet al.2009) are evidences of non- linearity effects generated by frequency modulation phenomena. At intermediate scales as for geotechnical applications, handling the interaction between the structure and the ground at the same time is crucial, particularly because of the potentially significant degra- dation of the shear modulus (Kramer1996; Del´epineet al.2009).

It is another reason why using non-linear stress–strain relations is interesting in seismic wave propagation modelling.

At local scales, non-linear extensions of standard elastodynamic equations have been suggested to better describe wave propagation in complex porous media. For instance, the poroelastic wave equations given in Biot (1956a,b) have been extended to non-linear poroelastic wave equations (Biot1973) in the particular context of Hertz–Mindlin rheology. Such approach has been applied to wave processes with strong acoustic non-linearity in porous rubber- or sandstone-like media mainly in one dimension (Ostrovsky1991) or to porous media containing spherical or cylindrical pores by Don- skoyet al.(1997) and to unconsolidated granular media by Dazel &

Tournat (2010). This theoretical framework has been developed in the acoustic community in an effort to understand non-linear mea- sured responses when the source amplitude is increased for instance in compact granular materials under gravity.

In 1-D, the interest of the geophysical community to non- linearities is obvious as 1-D non-linear response validations and ver- ifications using different numerical solutions have recently focused the attention on canonical/benchmark cases (R´egnieret al.2016a, 2018). Non-linear models in pure solid or poroelastic rheologies have been studied mainly by developing 1-D analytical solutions in the frequency domain in the context of dispersive granular media (Hokstad2004; Tournatet al.2003,2004; Tournat & Gusev2010;

Leglandet al.2012) or in the time domain for non-linear constitutive laws (Berjaminet al.2017) as well as 1-D finite differences (FD) in the time domain (Favrieet al.2014) or recent 2-D lattice approaches (Wallen & Boechler2017). In the studies of Tournat’s group, non- linear effects have been exhibited via second harmonics generated by, for instance, non-linear Hertzian contact rheologies or shear to longitudinal mode conversions in granular packed media. 1-D analytical models are essentially formulated using power law-based non-linear elastic and acoustic rheologies for granular and unconsolidated dispersive media. Power law degrees equal or lower than 2 are generally used and quadratic Hertz–Mindlin stress–strain relations are introduced with one non-linear parameter homogeneous to the shear or bulk modulus of granular media at the laboratory scale.

In Favrieet al.(2014) a 1-D non-linear version of the Zener model is proposed. The generalized Zener viscoelastic model degenerates correctly towards a pure non-linear elastic model when attenuation effects vanish. Besides, the non-linear elastic stress–strain relations under study are Lennard–Jones potential based functions or cubic functions like the Landau’s model which are widely used in non- destructive testing and slow dynamics modelling.

High order polynomial stress–strain functions are commonly introduced in the frequency domain to homogenize solid–solid or solid–fluid interactions from pore to microscale. Ostrovsky & John- son (2001) made a review of different non-linear elastic behaviours of earth materials and their impact in earth and material science (strong ground motion, non-destructive testing). They describe different sources of non-linearities in waves travelling through rocks due for instance to geometry of cracks and solid matrices, grain shapes, fluid content/saturation effects, the nature of soften/weak materials (presence of cracks, grain–grain or grain–fluid contacts, etc.), as well as hysteresis, relaxation or attenuation effects in these materials (slow dynamic effects). In many experiments, second and third harmonic spectrum amplitudes are evidence of non-linear behaviour and are varying according to the strain amplitude of the fundamental mode and with quadratic and cubic frequencies. In this context the authors give a theoretical high order polynomial expression of the stress–strain relation in which non-linear coefficients are appearing and can be modelled as combinations of the elastic moduli. The non-linear rheologies basically are derivations of the second potential invariant and can be expressed by power laws as quadratic or cubic stress–strain relations. The power laws (with power of 3/2, 2 or 3) of the stress–strain relation and the coefficients involved in the different terms of these laws are highly dependent on the shape of the pores or microcracks involved in the solid material (spherical, cylindrical, ellipsoidal, etc.; Nazarov

& Sutin1997; Nazarov2001; Ostrovsky & Johnson2001; Yu-Lin et al.2009). In this study, and without loss of generality, we will carry out the numerical verification using a quadratic term in the strain–stress relationship.

Concerning the numerical modelling of seismic wave propagation in realistic media, general overviews (e.g. Carcioneet al.2002;

Moczoet al.2011,2014) introduce most of the numerical methods dedicated to mechanical dynamics that include different direct methods such as pseudospectral, continuous (SEM)/discontinous

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(DG) Galerkin finite-element methods, and FD techniques in frequency or in time domains. 2-D or 3-D FD time domain (FDTD) approaches are commonly applied to discretize and solve the wave equations for elastic, viscoelastic, anisotropic and numerous dispersive media (Levander1988; Robertssonet al.1994; Graves1996, 1999; Bohlen2002; Saenger & Bohlen2004). These techniques have been generally applied using cartesian and staggered mesh discretizations. But more recently, more sophisticated FD techniques using high-order time-stepping and deformed meshes with non- staggered and collocated mesh point discretization (Zhang & Shen 2010; Zhanget al.2012; Sunet al.2018) have been introduced as well as non-deformed meshes using 2-D immersed methods (Lom- bardet al.2008; Chiavassa & Lombard2013; Blancet al.2014) with specific operators applied at physical interfaces and at the free surface (with mirror (Zhang & Chen2006; Zhanget al.2012) or non-centred schemes (Lombardet al.2008)). Also, recent develop- ments in including Zener body attenuation models have been made in the SEM (Blancet al.2016) for a constant quality factor.

Usually the linear viscoelastic behaviour is modelled by a memory variable formalism in order to ease the numerical implementation of the time convolution operator associated with the stress–

strain relation (Moczoet al.1997; Xu & McMechan1998; Hestholm 1999; Olsenet al.2000; Day1998; Day & Bradley2001; Wanget al.

2001; Bohlen2002; Saenger & Bohlen2004). The excessive memory storage requirements associated to this material-independent memory variable approach needed to be numerically optimized and reduced by using coarse sampling approach for instance (Kristek &

Moczo2003). The aim of this study is to avoid the use of memory variables by extending an auxiliary differential equation (ADE) approach in three dimensions for a simple (quadratic term) but repre- sentative non-linear behaviour. To our knowledge, for the first time in 3-D seismic wave propagation modelling, the ADE formalism is used for implementing a non-linear and viscoelastic constitutive law. Such a formulation has been used with success for a linear viscoelastic case by Dhemaiedet al.(2011) in an explicit FDTD-PML scheme using a second-order spatial and time stepping accuracy and a classic staggered grid for a single Zener body viscoelastic model.

The authors were inspired by an ADE electromagnetic application in ground penetrating radar by Rejibaet al.(2003) and were moti- vated by extending the ADE approach to the seismic wave equations for a desired viscoelastic model.

Attenuation is commonly described mathematically by different viscoelastic models defined as a combination of relaxation mech- anisms (series and parallel settings of springs and dashpots) as in Moczo & Kristek (2005) that allow different frequency dependent formulations of the attenuation. Maxwell, Kelvin–Voigt or Zener body models are generally used, and the reader can be referred to Semblat & Pecker (2009) for more details about those models. The Zener body model is used here and provides a simple stress–strain formulation in the frequency domain and is defined as a Kelvin–

Voigt cell and a spring in series.

The use of ADE formalism in 3-D-FDTD, for non-linear and dispersive material, is widely used for electromagnetic waves propagation and particularly for photonics and active plasmonics applications where Lorentz–Drude, Raman and Kerr non-linear terms are incorporated (Taflove & Hagness2005; Greene & Taflove2006;

Tafloveet al.2013). This approach is very uncommon (almost in- existant) in seismic wave propagation despite its great potential, except in rare modelling studies in 1-D dispersive and attenuated fluid/acoustic biophysical media (Jim´enez et al. 2016) or non- destructive testing (Lombard & Piraux2011). The only mention of ADE formalism concerns the implementation of ADE-perfectly

matched layer (PML) absorbing boundary conditions (Martinet al.

2010; Zhang & Shen2010; Moczoet al.2014). The main specificity, and probably the best asset of this method, is that it allows a clear separation to be established between the set of propagation equations: the strain–velocity equation, the velocity–stress equation, and the rheological model defined by the stress–strain relation. Then by conviniently using the differentiation theorem for the Fourier transform, an inverse Fourier transformation from the frequency to the time domain can be performed for each term of the stress–strain relation. As for all FDTD techniques the ADE approach rigorously enforces the vector field boundary conditions at interfaces of different media associated to a small fraction of the impinging pulse width : consequently it is an almost completely general approach that permits the modelling of a broad variety of dispersive and even non-linear materials. As a consequence, this method has particularly the advantage of allowing the flexible implementation of any dispersion law based on non-constantQquality factor law and on the complex expression of the desired viscoelastic modulus. More generally, by using the ADE approach, the stress–strain relations are not limited to expressions only based on linear dispersion laws or quasi-constant quality factors and can also incorporate non-linear rheological laws to take into account hysteretic, friction and dam- aging effects for instance.

In this study, we will explain how the introduction of a strain–

stress relation, involving non-linearity and attenuation, can be efficiently solved numerically in 3-D. We will present the numerical implementation and the numerical and physical verification of a simple Zener model associated with a quadratic non- linear term by introducing an ADE-FDTD approach. Our 3-D scheme includes a second order time-stepping and a fourth order staggered-grid spatial discretization to limit the numerical dispersion and convolutional (Komatitsch & Martin2007; Martin &

Komatitsch 2009; Martin et al. 2008, 2010; Bodetet al. 2014) or non-convolutional (Martinet al.2010) perfectly matched layer boundary conditions designed for elastic, viscoelastic or poroelastic seismic-wave modelling. We incorporate simultaneously these conditions along with viscoelastic properties inside the stress–strain formulations. It is worth noting that, in this study, non-convolutional PML using ADE formalism is implemented (Martinet al.2010) for more efficiency with less memory storage and subsequently faster computations.

In the following, we first give the governing equations of the non- linear constitutive law used. For sake of simplicity and without loss of generality for linear or non-linear stress–strain relations, we consider a homogeneous medium and an associated rheology defined by a single Zener body model for the viscoelastic part of the stress–

strain relation and an additional non-linear (quadratic) stress–strain relation term in one direction of space. The 3-D medium can thus be considered here as anisotropic in the direction of source polarization. We then show a set of 3-D numerical experiments to check for the accuracy and validity of the present numerical ADE-FDTD scheme, particularly concerning viscoelastic and non-linearity effects. The numerical experiments are designed for three different rheologies of the medium (elastic, viscoelastic and non-linear viscoelastic) to be compared if excited with the same source signal.

We concentrate our numerical (i.e. spectral) analysis and verification process on the isotropic non-linear viscoelastic rheology. The study of non-linear aspects is performed thanks to classical spectral and dispersion analysis tools. Different amplitude factors and frequency contents of the source excitation are tested to clearly exhibit the non-linear response of the medium. The influence of the source type on the non-linear effects is also discussed.

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2 G O V E R N I N G E Q UAT I O N S O F T H E L I N E A R A N D N O N L I N E A R

V I S C O E L A S T I C E Q UAT I O N S

In this study, we aim at directly introducing linear or nonlinear viscoelastic rheologies, but with a true constitutive law relating the strain deformations to the stresses by writing, in the frequency domain:

σ =

i=1,N

σi^L+σi^{N L}

=

i=1,N

aiεi+

i=1,N

j=1,M

bici jε_i^j, (1)

whereσi^Landσi^{N L}are, respectively, the linear and non-linear parts of the global effective tensor σ. ai are complex coefficients depending on the physical and geometrical properties of the linear viscoelastic part of the effective tensor. Subscriptirefers to one of the partial tensorsσi andjis thej-th polynomial degree of each term of the non-linear part of eachi-th partial tensorσi.b_iandc_ij are the polynomial coefficients of the non-linear part of the effective tensor. For simplicity, we will considerN=1 in all cases,M=1 in the linear case andM=2 in the non-linear case, the general effective tensor calculation being possibly treated in the same way. The strong form of the linear viscoelastic wave propagation equation in the time domain is given by:

ρ ∂²

∂t²U= ∇ ·σ,

(2) whereU= (ui)i=1,D (D being the space dimension) is the solid displacement vector andσ is the viscoelastic stress tensor. To introduce a viscoelastic rheology, the stress tensor is defined by a convolution product of the modulus tensor and the strain (Carcione 2007) as follows:

σi j =(C∗ε)i j (3)

εi j = 1 2

∂ui

∂xj

+∂uj

∂xi

, (4)

whereCis the modulus tensor of the viscoelastic solid matrix. In the frequency domain, by applying a Fourier Transform, we can write:

σˆi j =F(C∗ε)i j =( ˆCˆε)i j =λ^∗δi jεˆkk+2μ^∗εˆi j

εˆi j = 1 2

∂uˆi

∂xj

+∂uˆj

∂xi

, (5)

where indicesiand jcan be 1, 2 or 3 here in 3-D, and with the Einstein convention of implicit summation over a repeated index.

ε is the strain tensor of the viscoelastic solid matrix, and ρ is the density of the solid material.μ^∗=μ(¹⁺₁₊ⁱ^ωτ_i_ωτ⁰^γ^μ

0 ) is a complex shear modulus andλ^∗=λ(¹⁺ⁱ_1+iωτ^ωτ⁰^γ^λ

0 ) is a complex Lam´e coefficient parameter of the solid matrix. We can notice thatμ∗andλ∗are non-linear functions of the frequencyω.

After some algebraic manipulations, we can also compare the numerical and the analytical compressible and shear moduli of the viscoelastic medium, the expressions of the analytical moduli being given byM_P=λ∗ +2μ∗andM_S=μ∗and the expressions of the numerical moduli by

MP = σˆzz( ˆεzz+εˆx x)+σˆyy( ˆεyy+εˆx x)

εˆzz( ˆεzz+εˆx x)+εˆyy( ˆεyy+εˆx x) (6)

MS = σˆyz

εˆyz

. (7)

These expressions will allow us in the next section to validate the numerical solutions at least for the linear viscoelastic case.

For simplicity, but without loss of generality, we add non-linear behaviour to the stress–strain law and consider that the non-linearity is only present in one direction of space (the longitudinalzdirection for instance). We make the assumption that the fluid part is missing in the non-linear (quadratic) poroelastic formulation of Donskoy et al.(1997) or Dazel & Tournat (2010) and that the elastic part is replaced by the Zener linear viscoelastic strain–stress relation of Dhemaied et al. (2011). The stress–strain ˆσ(ω)=M(ω)ˆε(ω) relation in the frequency domain is given by:

σˆ = (λP,S+λ˜P,Siω)ˆε+Hεˆ^m 1+τ0iω ,

(8) whereHis a non-linear tensor, mis an exponent that generally depends on the nature of the non-linear rheology (due to ellipti- cal/spherical cracks, inclusions or cavities distributions, ...),τ0is the stress relaxation time,λP,Sand ˜λP,Sare, respectively, the unre- laxed (real) and relaxed (imaginary) parts of the Lam´e parameters.

Once these rheological laws are given in the frequency domain, they are expressed in the time domain after applying the inverse Fourier transform to each term of the stress–strain relation, and consequently the stress–strain equation yields:

σ+τ0∂tσ =λP,Sε+λ˜P,S∂tε+Hε^m,

(9) In a less compact form, the system of equations is developed as:

σi j+τ0

∂σi j

∂t =λδi jεkk+2μεi j+λδ˜ i j∂tεkk+2 ˜μ∂tεi j+Hi jδi jεi j^m

εi j = 1 2

∂ui

∂xj

+∂uj

∂xi

∂tεi j = 1 2

∂vi

∂xj

+∂vj

∂xi

, (10)

wherevi(i=1, 3 in 3-D) are the solid velocity vector components.

As stated earlier, we assume thatHxx=Hyy=Hxy=Hxz=Hyz=0 andHzz=Dwhere the parameterDcan take zero or non-zero value.

In the case ofD=0 we retrieve the linear isotropic viscoelastic case and forD=0 we retrieve a non-linear anisotropic viscoelastic stress tensor in the frequency domain with a non-linear anisotropy in one direction of space. A powerm=2 is chosen now in this study for the additional non-linear term without loss of generality. The relaxed Lam´e coefficients ˜λand ˜μare expressed as follows:

λ˜ =τ0λγλ

μ˜ =τ0μγμ (11)

with γλ= 1

λ τp

τ0

(λ+2μ)−2μγμ

γμ = τs

τ0

, (12)

whereτp andτs are thePandS-wave relaxation times. Eq. (10) is the required ADE defining the stress tensorσ(t) in the time domain. It has been easily implemented in an FDTD code using a semi-implicit scheme centred at time stepn+1 wherein yet-to-be

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Elements size:

1.4 × 1.4 × 1.4 mm x y z

1 1 1O

Point source implemented along the Oz axis (x_s= y_s= 0 ; z_s= 232.4 mm) Record line:

z_min= 232.4 mm Δz = 1.4 mm z_max= 252 mm

2000 grid points

200 grid points

(a) (b)

0

Not to scale

0 0.0025 0.005

Time (s) -6

-4 -2 0 2 4 6

A0 (A @ zero offset)

x 10-11

0 5000 10000

Frequency (Hz) 0

1 2 3 4

A0 (A @ zero offset)

x 10-8

Implemented source amplitude factor ^{x 1.5}^{x 1.25} x 1.0 x 2.0 x 1.75 x 2.25

x 0.75 x 0.5 x 0.25

{

Max. recorded amplitude

zero offset 2λ_P 3λ_P

5λ_P 4λ_P

Figure 1.Simplified illustration of our numerical setup (a). A dominant frequency of 1500 Hz for a Ricker wavelet point-force source has been implemented in the longitudinal (z) direction with several excitation amplitudes in the non-linear viscoelastic case (black diamonds correspond to picked maxima) (b).

0

0.005

0.010

0.015

Time (s)

0 Offset (m)1

elastic

Amplitude

2 0 Offset (m)1 2 0 1 2

linear viscoelastic

Offset (m)

non-linear viscoelastic

-1.0 -0.5 0 0.5 1.0x10 -14

0

0.5

1.0 x104

Frequency (Hz)

0 0.5 1.0 1.5 2.0 2.5

x10 -12

elastic linear viscoelastic non-linear viscoelastic

(a)

(b) ⁰ ^{Offset (m)}¹ ² ⁰ ^{Offset (m)}¹ ² ⁰ ^{Offset (m)}¹ ²

P

S

P

S

P

S

Figure 2.Seismograms (a) and spectrograms (b) of vertical component particle velocity (raw amplitude) recorded along the simulated models for: elastic, linear viscoelastic and non-linear viscoelastic rheologies medium (source excitation amplitude×1).

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computed velocity and displacement fields at timen+1/2 are used to create an update formula for a field known at time-step n; almost exactly as detailed in Dhemaiedet al.(2011) for the 2-D case. This ADE-FDTD integration scheme is detailed in the next section.

3 N U M E R I C A L E X P E R I M E N T 3.1 Setup

A second order displacement formulation is chosen and a fourth order staggered-grid FD spatial discretization in space (Graves1996) as well as a second order time-stepping algorithm are used to solve the whole set of equations. The ADE system of equations is discretized in time at the second order using a semi-implicit scheme as follows:

ρ vi^n+1/2=ρ vi^n−1/2+t(∂jσi jⁿ+Fi)

uⁿ_i⁺¹^/²=uⁿ_i⁻¹^/²+tv_iⁿ⁺¹^/²+t²viⁿ⁺¹^/²−vⁿi⁻¹^/²

t εⁿi j⁺¹^/²= 1

2

∂uⁿ_i⁺¹^/²

∂xj

+ ∂uⁿj⁺¹^/²

∂xi

∂tεⁿi j⁺¹^/²= 1 2

∂vⁿi⁺¹^/²

∂xj

+∂vⁿj⁺¹^/²

∂xi

σi jⁿ⁺¹(τ0/t+0.5)=σi jⁿ(τ0/t−0.5)+λδi jεⁿkk⁺¹^/²+2μεⁿi j⁺¹^/²

= +λδ˜ i j∂tεkkⁿ⁺¹^/²+2 ˜μ∂tεi jⁿ⁺¹^/²

= +Hi jδi jεi j² |^n+1/2, (13)

whereui and vi (i=1, 3 in 3-D) are the, respectively the solid displacement and velocity vector components.

We implement the following FD operator of each gradient of a variablefin a given directionx, similar operators being applied in the other two directionsyandz:

∂f

∂x |i+1/2,j,k≡ −fi+1,j,k+27fi,j,k−27fi−1,j,k+ fi−2,j,k

24 . (14)

Optimized convolution (Komatitsch & Martin 2007) or non- convolution (Martinet al.2010) frequency shift perfectly matched layers absorbing boundary conditions complete the implementation to efficiently truncate the computational domain and mimic this way an infinite medium. The advantage of this non-convolution PML is that we have more flexibility to implement different time-stepping schemes when compared to CPML formulation.

For the sake of clarity, we use here, respectively, second and fourth order discretization operators in time and space to inte- grate the equations as well as CPML boundary conditions. Be- sides, without loss of generality, higher order operators can also be used to solve the elastodynamic equations and add non-convolution PML conditions as in Martinet al.(2010), where we introduced eighth order Holberg space operators and fourth-order Runge–Kutta (RK4) time-stepping integration. The discretization could also be replaced by more sophisticated space discretizations in space and time on deformed meshes and collocated grids as those mentionned in the introduction. But here, the fourth-order staggered grid numerical discretization and second-order time-stepping we choose are enough for the purpose of this study because we just want to exhibit attenuation and non-linear effects in 3-D with reasonable accuracy. In addition, it should be emphasized that, to regularize possible sharp shocks related to the non-linearity introduced, we use non-centred fourth-order spatial discretization of the quadratic

termε²involved in the strain–stress law and applied in the source polarization directionz only (here only the componentHzz = D is non-zero in theHnon-linear tensor, andεzz is denoted byεto alleviate notations). To avoid possible high order spurious modes, we denote byεi,j,k+1/2=0.5(εi^L,j,k+εi^R,j,k) the average strain computed at a cell face discontinuity where two strain deformation states L (left) and R (right) are defined. We assume thatεi²,j,k+1/2= εi,j,k+1/2εi^L,j,k+1/2ifεi,j,k+1/2>0, andεi²,j,k+1/2=εi,j,k+1/2εi^R,j,k+1/2

if εi,j,k+1/2<0. Here we choose ε_i,j,k+1/2/^L =εi,j,k and εi^R,j,k+1/2=εi,j,k+1.

A simplified illustration of our numerical setup is presented on Fig.1(a). In all cases, we perform our simulations using a dominant frequency of 1500 Hz for a Ricker wavelet point-force source, implemented in the longitudinal (z) direction (see Figs1a and b).

We have sampled the spatial and time steps in our simulations in order to have access to a wide range of wavelengths compared to the source signal. The time step is thus chosen ast=0.4μs and the spatial step isx=y=z=0.0014 m. The grid mesh is discretized by 1800 points along the longitudinalz-axis and, 200 points over the traversalxdirection and 200 points over the depth y. A source is located atz=0.2324 m from the left boundary of the thin slice and receivers are located at all discretization points along the longitudinal length and in the middle of the medium (the length of the models along the longitudinal axiszis 2.52 m). We checked the accuracy of the numerical method for pure viscoelastic and non-linear viscoelastic cases for two different sources (a Ricker and a Gaussian-modulated sine pulse source) that will be used in the next sections of this study. For sake of clarity, the reader is referred to the appendix section for more details. In that section P-andS-wave velocities are very well reproduced for both types of sources in the viscoelastic case. Furthermore, in the non-linear viscoelastic case, theσzzcomponent of the stress tensor (computed using the non-centred discretization of the non-liner term at the cell faces described before) is well reproduced in the frequency domain.

3.2 Numerical verification of the nonlinear viscoelastic regime

3.2.1 Classical dispersion analysis using a wideband pulse Now, we focus our attention to the physical aspects of our study. To illustrate the effect of the current non-linear regime (Fig.2a), we first present the seismograms obtained for a reference elastic and homogeneous medium characterized by a pressure-wave velocity (Vp) of 245.63 m s⁻¹, a shear-wave velocity (Vs) of 122.65 m s⁻¹ and a bulk densityρ=1610 kg m⁻³, these values corresponding to experimental observations in unconsolidated granular media recently studied in Bodetet al.(2014). A second simulation consid- ers the viscoelastic case, based on similar homogeneous properties but with stress and strain relaxation times, respectively, set toτ0

=0.28 ms andτp=τs=0.3 ms. A third simulation finally presents the non-linear case. TheDparameter described earlier is set to 5×10¹³×ρV_p²to highlight non-linear regime effects.

The seismograms of particle velocity recorded in thezdirection are given as a function of offset (source-recording point distance along the longitudinalz-axis) and time (presented on Fig.2a). They consist in 1634 traces of 50 000 amplitude samples. Except for vis- ible variations in amplitude of particle velocity between the elastic case and the two viscoelastic cases, distance–time data systemati- cally depict two distinct events: aP- and aS-wave train (PandS

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0 2000 4000 6000 8000 10000

100 150 200 250 300 350 400

Frequency (Hz)

Phase vel. (m/s)

P-elast. picked P-elast. analytical S-elast. picked S-elast. analytical P-visco. picked P-visco. analytical S-visco. picked S-visco analytical P-non lin. visco. picked S-non lin. visco. picked

Figure 3.Dispersion curves: maxima picked from frequency-phase velocity transforms of seismograms for bothPandSwaves compared to corresponding theoretical dispersion curves.

Figure 4. Seismograms of particle velocity (raw amplitude) recorded along the simulated models for different source excitation amplitudes.S-wave muted.

on Fig.2a) of similar apparent velocities for each rheology. How- ever, the effect of viscoelasticity is clearly apparent on corresponding spectrograms (Fig.2b) which present important differences in terms of amplitude and shape. The non-linear case typically presents frequency-up conversions with at least two amplifications at higher frequencies that are linked to the source peak frequency (1.5 kHz)

but could not be exactly and accurately associated to a specific harmonic: the fundamental mode is located around a frequency of 1900 Hz, and the third harmonic is roughly 5700 Hz, but the second harmonic that should be located around a frequency of 3800 Hz does not appear clearly. In addition, the wavefield recorded for each case has been transformed into the frequency-phase velocity domain (by

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Figure 5. Amplitude spectrum of particle velocity (raw amplitude) recorded along the simulated models for different source excitation amplitudes.S-wave muted.

a slant-stack in common shot gathers after correction for geometrical spreading) to quantify the phase velocities of theP- andS-wave events cited above. The resulting dispersion curves, presented on Fig.3clearly show that the simulations perfectly describe implemented phase velocities: constant velocity in the elastic case and, by definition, dispersive relationship between velocity and frequency in the viscoelastic case.

3.2.2 P-wave extraction using advanced preprocessing tools and pulse modulation

We will consider the Ricker source described earlier as a reference in term of amplitude. More runs involving exactly the same parameters as the latter non-linear viscoelastic model are presented, but with different amplitudes of the source (from 0.25 ×to 2.25 ×the excitation amplitude of the first case, as shown on Fig.1b). We will focus on the longitudinalzdirection, and consequently on theP- wave train only. The seismograms, presented on Fig.4obtained for each source excitation amplitude, are thus muted as soon asP- and S-wave trains can be separated. Each corresponding spectrogram on Fig.5clearly depicts the harmonics observed earlier resulting from implemented non-linearities.

In order to clearly identify the non-linear effects, we extracted traces from each seismogram and spectrogram along the line at four different offsets, corresponding to two to five times the dominant wavelength of the source (λP ≈0.18 m). These traces are presented

on top of Fig.6. They correspond to the signals computed at each different source excitation amplitude (from×0.25 to×2.25 the amplitude of the reference run presented in Section 3.2.1; see colour- scale on Fig.1b). The corresponding spectra are presented on top of Fig.7, on which the main frequency peakA1 and two other peaksAandAclearly appear (maxima are picked and shown as black plus signs). These maxima are not in ratio 2 in frequency, but seem to correspond to the third and fifth harmonics, respectively, which poses the question of their correspondence with harmonic wave components. These traces are then normalized to the source maximum amplitude in both time and frequency domains (bottom lines of Figs6and 7). When the normalization in time remains equivalent (every normalized traces can be considered superimposed on Fig.7), the ratio presented in frequency clearly shows opposite behaviour as a function of source amplitude depending on the considered peak. The black arrows, given on top of Fig.7as an example, depict a non-monotonous behaviour of these harmonics as a function of source amplitude.

We picked the amplitude of each peak and presented, on Fig.8, their ratios between the first peak and the other peaks. These spectral ratios were computed at the 4 different offsets corresponding to two to five times the dominant wavelength of the source (λP ≈ 0.18 m) described above. The colours on Fig.8depend on these offsets and ratio are given of course for every source excitation amplitudes (from×0.25 to×2.25 the amplitude of the reference run presented in Section 3.2.1). At far-field offsets, the ratio shows

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Figure 6.Traces recorded at four different offsets given as a function of the source-signal dominant wavelength (λP≈0.18 m) presented for each different source excitation amplitude (from×0.25 to×2.25; see colour-scale on Fig.1b) with corresponding normalization to the amplitude at the source (A0). A is the amplitude of the trace at corresponding offset.

strong variations with source amplitude and tends to remain stable at near-field offsets. As stated earlier, picked maxima are not in ratio 2 in frequency which suggests overlap (interferences) between adjacent harmonics due to the wide frequency band of the Ricker source and dispersion effects (a 1900 Hz fundamental frequency and a 5700 Hz third harmonic frequency). This observation illustrates as well the fact that non-linear effects accumulate over the distance to the source, as these ratios increase at fixed amplitudeA1 and increasing distance.

Consequently, to exhibit and individualize the different peaks more precisely, we introduce a narrow frequency band pulse source implemented as a Gaussian-modulated sine pulse with a dominant frequency of 1500 Hz. Fig.9compares normalized traces obtained from both the Ricker pulse (with amplitude of the source 2×the excitation amplitude of the first case (3.2.1)) and the Gaussian- modulated sine pulse (Fig.10). From these seismograms (Figs10a and b), we computed amplitude (Figs10c and d) andf−kspectra (Figs10e and f). The spectra clearly show that, within the same propagation ranges and domains, the source modulation allows the identification of all consecutive harmonics (frequency-up conversions in ratio 2 in frequency), corresponding to the expected non-linear effect. In addition, for example in the case of strong non-linearities arising from a high amplitude of the source, the su- perimposition of all spectra corresponding to all successive offsets along the analysed direction (Fig.11) highlights the overlapping process associated with wideband pulse. In the present case, all first

six successive modes are clearly defined in terms of peak frequency location and amplitude. In this case of a Gaussian-modulated sine pulse, we can observe that the amplitude maximaA1 have similar non-linear effects (Figs12a and b) as theA1 maxima in the Ricker wavelet source case (Figs 11a and b), and are much better indi- vidualized on a wider frequency range. However, the spectral ratio amplitudesA3/A1³have almost the same exponentially increasing patterns asA/A1³in the Ricker source case at almost similar offsets. In Fig.11(b), the second and third peaksAandAare well separated while in Fig. 12(b) the maximaA2 and A3 are almost superimposed. The ratios of second and third harmonic amplitudes A2 andA3 over first peaksA1,A1² andA1³ present a more pro- nounced non-linear behaviour (Figs12b–d) at offsets closer to the source location when compared to the Ricker wavelet source case.

More particularly, the ratios of second and third harmonic ampli- tudesA2 andA3 overA1² andA1³ show an increasing trend with offset (Figs12c and). Small oscillations of those spectral ratios for the gaussian-modulated sine pulse appear at small offsets probably due to near field effects, and seem to be mainly associated with amplitude oscillations ofA1. Finally, in both non-linear cases, we can see that the spectral ratios are increasing (Figs11e and f,12e and f) and are reaching similar high values for values ofA1 close to zero at very far offsets to the source while they are converging asymptotically to a constant value for high values ofA1 close to the source.

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Figure 7.Amplitude spectra and spectral ratio of traces recorded at four different offsets given as a function of the source-signal dominant wavelength (λP ≈0.18 m) presented for each different source excitation amplitude (from×0.25 to×2.25; see colour-scale on Fig.1b).Ais the amplitude of the trace at corresponding offset, A0 the amplitude at source andA1,AandAare the amplitudes of the three first maxima detected in the spectrum at each offset. The black arrows are just given on top as an example showing their non-monotonous behaviour as a function of source amplitude.

(a)

(b)

Figure 8. Spectral ratio computed at different offsets for every Ricker wavelet source excitation amplitudes (the four different offsets, corresponding to two to five times the dominant wavelength of the source (λP≈0.18 m), are given with colours depending offset from blue (2λP) to green (5λP)).

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0 0.005 0.01 0.015 Time (s)

-2 0 2

A0

0 2000 4000 6000 8000 10000

Frequency (Hz) 0

0.5 1

A0

0 0.01

Time (s) -1

-0.5 0 0.5 1

0 5000 10000 Frequency (Hz) 0

0.2 0.4 0.6 0.8 1

(a) (b)

A @ 2λP

Gaussian-modulated sinusoidal pulse Ricker pulse

Gaussian-modulated sinusoidal pulse (implemented) // // // // (recorded) Ricker pulse (implemented)

// // (recorded)

Figure 9.Test comparing normalized traces obtained from the Ricker pulse and a gaussian-modulated sine pulse: (a) at the source location and (b) at near offset (2λP).

3.3 Discussion on non-linear regimes using non-dimensional analysis

In order to characterize better the regime (linear, weakly or strongly non-linear) under which the numerical experiments studied in the previous section are performed, we consider below the simplest non- linear model and its associated non-dimensional numbers. The simulated problem corresponds to an oscillating point (dipolar) source along thez-direction in a homogeneous elastic solid medium, showing linear viscoelastic isotropy. The non-linearity, as described earlier, is anisotropic (only acting on theσzz and εzz relation) and quadratic. The non-linear propagation is inspected along the z- direction. To our knowledge, there is no analytical solution in this configuration for the non-linear process of harmonic generation. We therefore recall here the usual parameters for plane wave propagation of longitudinal waves in a homogeneous medium with quadratic non-linearity. First, we define linear and non-linear time or length scales denoted byτa,τnl,La,Lnl.Laandτacorrespond to the characteristic length and timescales over which the initial amplitude of a wave is divided byedue to dissipation effects, andLnlandτnl

are the characteristic non-linear length and timescales over which non-linear effects develop. If the receivers are located close to the source then non-linearities cannot accumulate enough. If they are at a distance to the source lower thanLnlthen the non-linear effects can be observed but are still in a regime of accumulation. If it is equal toLnlthen the non-linearities are fully accumulated and developped andLnldenotes the shock front distance. If we non-dimensionalize the non-linear equations, they can be expressed as:

ω²ρUoU=1/Ladiv((λP,S+λ˜P,Siω)Uo/Laεˆ

1+τ0iω +DU_o²/L²_aεˆ²) U= (λP,S+λ˜P,Siω)Uo/La²

ω²ρU_o(1+τ0iω) div( ˆε)+ DU_o²

ρω²U_oL³_adiv( ˆε²) (15) whereU=U×Uo,xi =x_i×La,t=t/ω,ε=ε×Uo/Lawithε

=0.5(∇U+ ∇U^T). If we denote the modulus of the viscoelastic seismic velocity byV pa=(^||^M_ρ^P^||)¹^/², where MP is the complex modulus associated to thePwave, then the (visco-)elastic length scale can be considered homogeneous to the seismic wavelengthLa

=Vpa∗T=Vpa/ω. Then the previous equations can be simplified as:

U = ρV p_a²Uoω²/V p²_a ω²ρUo

div( ˆε)+ DU_o²La

ρω²UoL⁴_adiv( ˆε²)

=div( ˆε)+ DU_o²Laω⁴ ρω²UoV p_a⁴div( ˆε²)

=div( ˆε)+ DUoLaω² ρV p_a⁴ div( ˆε²)

=div( ˆε)+ La

Lnl

div( ˆε²)

=div( ˆε)+div( ˆε²), (16)

where we can consider the non-dimensional Goldberg number that measures the ratio between the linear attenuation effects and the non-linear elastic effects.can be expressed as

= τa

τnl

= La

Lnl

, (17)

where the different lengths and timescales can be formulated as:

Lnl = ρV p⁴_a ω²DUo

τnl = Lnl

V pa

= ρV p³a

ω²DUo

τa =τnl

La =V pa/ω La =Lnl

τa

τnl

. (18)

Since we have D=DρV_p² withD =5.10¹³ we can compute :

= DUoLaω² V p²_a = D V_o

V p_a. (19)